Choice of computational method for swimming and pumping with nonslender helical filaments at low Reynolds number

Abstract

The flows induced by biological and artificial helical filaments are important to many possible applications including microscale swimming and pumping. Microscale helices can span a wide range of geometries, from thin bacterial flagella to thick helical bacterial cell bodies. While the proper choice of numerical method is critical for obtaining accurate results, there is little guidance about which method is optimal for a specified filament geometry. Here, using two physical scenarios-a swimmer with a head and a pump-we establish guidelines for the choice of numerical method based on helical radius, pitch, and filament thickness. For a range of helical geometries that encompass most natural and artificial helices, we create benchmark results using a surface distribution of regularized Stokeslets and then evaluate the accuracy of resistive force theory, slender body theory, and a centerline distribution of regularized Stokeslets. For the centerline distribution of regularized Stokeslets or slender body theory, we tabulate appropriate blob size and Stokeslet spacing or segment length, respectively, for each geometry studied. Finally, taking the computational cost of each method into account, we present the optimal choice of numerical method for each filament geometry as a guideline for future investigations involving filament-induced flows.

Helical and rod-shaped bacteria swim in helical trajectories with little additional propulsion from helical shape

“…However, thrust and drag can be estimated from a model that ignores the hydrodynamic interactions between the cell body and flagellar bundle. We calculate the force and torque on the body and flagellar bundle separately using the RSM to calculate resistance matrices that express the forces and torques in terms of their linear and rotational velocities ( 48 ). Imposing the kinematic constraint of a fixed bundle-cell orientation and net force and torque balance yields swimming velocities and body rotations that are qualitatively in agreement to the full RSM calculations (see the Supplementary Materials for details).…”

Effects of cell morphology and attachment to a surface on the hydrodynamic performance of unicellular choanoflagellates

“…We use the regularized Stokeslet method in three dimensions to compute the fluid flow due to the choanoflagellate's undulatory flagellum [28]. Forces are distributed on the surface of the cell body at discrete material points [29] as well as at discrete points along the centrelines of the flagellum and each of the microvilli [30]. The force density concentrated at one of these points x k is:…”

Mixing and pumping by pairs of helices in a viscous fluid

“…where p is fluid pressure, u is fluid velocity, µ is fluid viscosity, and F(x) is a force density representing the force of the helices on the fluid. Following our previous model of a fluid-helix system [19], we choose a regularized Stokeslet formulation [7] where forces are distributed along the centerline of the helices [20]. Rather than assuming that these are pointforces, the force density concentrated at a point x k is assumed to be…”

“…The regularized Stokeslet method has been used to simulate flows at the microscale in applications including hyperactivated sperm motility [21], nodal cilia flow in embryology [23], optimal cilia design [24] and synchronization of waving elastic filaments [25]. The validation of this method using theory as well as comparison of results with those of other numerical methods or experiments has also been addressed, for example [7,20,26]. Figure 1 shows the configuration of two upright helices whose axes (of length L) are parallel, placed a distance d apart.…”